- Open Access
Finding multiple core-periphery pairs in networks
Phys. Rev. E 96, 052313 – Published 22 November, 2017
DOI: https://doi.org/10.1103/PhysRevE.96.052313
Abstract
With a core-periphery structure of networks, core nodes are densely interconnected, peripheral nodes are connected to core nodes to different extents, and peripheral nodes are sparsely interconnected. Core-periphery structure composed of a single core and periphery has been identified for various networks. However, analogous to the observation that many empirical networks are composed of densely interconnected groups of nodes, i.e., communities, a network may be better regarded as a collection of multiple cores and peripheries. We propose a scalable algorithm to detect multiple nonoverlapping groups of core-periphery structure in a network. We illustrate our algorithm using synthesized and empirical networks. For example, we find distinct core-periphery pairs with different political leanings in a network of political blogs and separation between international and domestic subnetworks of airports in some single countries in a worldwide airport network.
Physics Subject Headings (PhySH)
Article Text
Many complex systems can be expressed as networks in which a node represents an object (e.g., person, web page, protein) and an edge represents the relationship between two objects (e.g., friendship, hyperlink, physical interaction). A network can be characterized by microscale, mesoscale, and macroscale structural patterns such as the degree (i.e., the number of edges that a node has), clustering coefficient, and diameter . Among various structural properties of networks, community structure is a representative mesoscale structure of networks . A community is a group of nodes that are densely interconnected and sparsely connected to nodes in different communities. Nodes in the same community often share a role (for an exceptional case, see Ref. ), and therefore identifying communities aids classification of nodes and visualization of networks .
Core-periphery structure is another mesoscale structure of networks, with which we view a network as being composed of two groups of nodes called the core and periphery. Although the definition varies, a core is often defined as a group of densely interconnected nodes and a periphery as a group of nodes that are densely connected (i.e., adjacent) to the core nodes but not to other peripheral nodes . A core and community are both groups of densely interconnected nodes but have a difference; a core connects densely to its periphery, whereas a community is not densely connected to other nodes outside it. Core-periphery structure has been found in various networks, including brain networks , metabolic networks , protein interaction networks , social networks , international trade networks , financial networks , and transportation networks . For example, in a coauthorship network among researchers, leading researchers often publish papers with other leading researchers, forming a core, while other researchers tend to publish papers with particular leading researchers such as those in the same research group, forming a periphery .
Borgatti and Everett introduced the first quantitative formulation of core-periphery structure . In the discrete version of core-periphery structure, which we will focus on in this paper, they introduced an idealized core-periphery structure in which core nodes are adjacent to all other nodes, and peripheral nodes are adjacent to all core nodes but not to any peripheral nodes. Although it is also realistic to assume that the core-periphery connectivity is sparser than the core-core connectivity , we will focus only on the idealized core-periphery structure in the present study. Borgatti and Everett sought for the assignment of all nodes in a given network to a core or periphery such that the network is as close as possible to an idealized core-periphery structure. Following their study, many core-periphery detection algorithms have been developed . These algorithms aim to identify a single core-periphery pair embedded in a network. However, a network may be better regarded as a collection of multiple core-periphery pairs , which is the focus of the present study. For example, coauthorship networks would be composed of multiple groups of researchers. Researchers would often collaborate with the leading researchers in the same group but not with other researchers in the same group, which may lead to core-periphery structure within the group . Previous studies in this direction have not provided a tailored scalable algorithm to this end. A study focused on a related but different type of multiple core-periphery structure . Other algorithms aim to detect multiple cores but do not assume that peripheral nodes are sparsely connected to each other . A network can also have multiple disjoint cores in the form of
We present a scalable algorithm to detect multiple nonoverlapping core-periphery pairs in networks, each of which is as close as possible to an idealized core-periphery structure. Our algorithm automatically determines the number and the size of the core-periphery pairs. Various algorithms to detect core-periphery structure in networks are classified as density-based and transport-based algorithms . Densely-based algorithms posit that a core is a densely connected group of nodes, whereas transport-based algorithms posit that a core is a group of nodes that can be reached from other nodes along short paths. In the present study, we focus on density-based algorithms.
We extend the idealized core-periphery structure introduced by Borgatti and Everett to the case of multiple pairs of a core and a periphery. In the Borgatti-Everett (BE) algorithm, one considers a graph (i.e., network) composed of
The discrete version of the BE algorithm, which we consider in the present study, seeks
We extend the idealized core-periphery structure to the case of multiple core-periphery pairs. Let
where
We seek
where
We use a label switching heuristic to maximize
The increment in
where
A detected core-periphery structure may be statistically insignificant . Therefore, we adapt a statistical test in the case of a single core-periphery pair to the case of multiple core-periphery pairs.
In the statistical test for a single core-periphery pair , we measure the significance of a core-periphery pair by a quality function based on the Pearson correlation coefficient , which is defined by
where
In the case of multiple core-periphery pairs, we apply essentially the same statistical test to each core-periphery pair detected in the original network. For each detected core-periphery pair, we first calculate
If we test
We have decided to use
For the synthetic networks with planted core-periphery structure, we measure the difference between the true core-periphery structure
where
We compare the proposed algorithm with the BE algorithm, which detects a single core-periphery pair by maximizing We compare the performance of the three algorithms on four different types of synthetic networks with a planted core-periphery structure schematically shown in Fig. . We generate the synthetic networks using stochastic block models . We draw label Schematic illustrations of the adjacency matrices of the networks generated by stochastic block models. The filled blocks correspond to the entries that are equal to 1 with probability As a first example, we consider a network composed of a single core-periphery pair [Fig. ]. We set The VI values between the true and inferred core-periphery structure for the three algorithms. Rows S1, S2, S3, and S4 correspond to the networks with planted core-periphery structure shown in Figs. , , , and , respectively. The color of each cell indicates the VI value. The white cells are those for which we did not calculate the VI values, i.e., we only computed them for As a second example, we examine networks composed of two core-periphery pairs [Fig. ]. We set In empirical networks, there may be nodes that are better regarded not to belong to any core or periphery. Therefore, as a third example, we consider a network composed of a single core-periphery pair and residual nodes [Fig. ]. We regard the block of the residual nodes as a single group of nodes, like a core or periphery, when calculating the VI value. Let As a fourth example, we consider networks composed of two core-periphery pairs and residual nodes [Fig. ]. We set We apply the three algorithms to three empirical networks. For directed and weighted networks, we disregard the direction and the weight of edges. Consider the karate club network , which has The core-periphery structure detected by the three algorithms is shown in Fig. . The BE-KL algorithm detects a single core-periphery pair such that both the instructor and president are core nodes [Fig. ], neglecting the fissure of the club. The two-step algorithm detects two core-periphery pairs, each of which consists mostly of the members with the same leanings [Fig. ]. In particular, the instructor and the president belong to the core of the different core-periphery pairs. Two neutral members, nodes 10 and 19, are assigned to the president's core-periphery pair, which does not agree with the self-reports by the members. The residual nodes consist of the members on the instructor's side, those on the president's side and a neutral member. Our algorithm detects almost the same two core-periphery pairs as that detected by the two-step algorithm [Fig. ]. Core-periphery structure of the karate club network detected by (a) the BE-KL algorithm, (b) the two-step algorithm, and (c) our algorithm. The filled and blank cells indicate Next, we compare the density of edges within core-periphery pairs. For each significant core-periphery pair, we compute the density of edges within the core, that of edges within the periphery and that of edges between the core and periphery. Then, we average each type of edge density over all significant core-periphery pairs (without weighing by the size of core-periphery pair when calculating the average). We show the edge densities for the karate club network in Fig. . For all algorithms, the average density of intracore edges and that of core-periphery edges (i.e., edges connecting a core node and a peripheral node) are larger than the edge density for the entire network, The second example is a political blog network , which has The core-periphery structure detected by the three algorithms is shown in Fig. . The unique core detected by the BE–KL algorithm is a mixture of liberal and conservative blogs [Fig. ]. The peripheral blogs are mostly adjacent to blogs with the same political leaning. However, the structure detected by the BE-KL algorithm alone does not tell this unless we refer to the political leaning of the individual blogs. A different algorithm for a single core-periphery structure yielded similar results for the same network . The two-step algorithm detects three core-periphery pairs, each of which mostly comprises the blogs with the same political leanings [Fig. ]. Two core-periphery pairs are much larger than the third one and have the opposite political leanings. The third small core-periphery pair is mainly composed of liberal blogs. In each core-periphery pair, a majority of the peripheral nodes is densely interconnected, which is against the idealized core-periphery structure. This is due to the community detection step that partitions a network into communities with dense intracommunity edges. In fact, the average density of intra-peripheral edges within a core-periphery pair is Our algorithm detects two core-periphery pairs, each of which mostly comprises the blogs with the same political leaning [Fig. ]. The detected two core-periphery pairs are smaller than those detected by the two-step algorithm. More nodes are classified as residual nodes than by the two-step algorithm. The average density of intraperipheral edges within a core-periphery pair is Our third example is a network of airports, which has Figure shows the core-periphery structure detected by the three algorithms. The BE-KL algorithm detects a dense core composed of 89 airports scattered in different geographical regions [Fig. ]. The peripheral airports are rarely adjacent to the core airports in other regions. Furthermore, the peripheral airports tend to be adjacent to other peripheral airports in the same region, which is inconsistent with the notion of the periphery. The two-step algorithm detects 16 geographically concentrated core-periphery pairs [Fig. , in which some peripheral airports are densely interconnected within the core-periphery pairs. The average density of intraperipheral edges within a core-periphery pair is 0.0383, which is approximately 10 times larger than the edge density for the entire network, Our algorithm detects 10 geographically concentrated core-periphery pairs [Fig. ]. The partition of the worldwide airport network into geographically distinct groups of airports found here is consistent with the previous results derived with community detection algorithms . Compared to the two-step algorithm, the peripheral airports detected by our algorithm are not densely interconnected; the average density of intraperipheral edges within a core-periphery pair is 0.000073, which is smaller than the edge density for the entire network, We further analyze the core-periphery structure obtained by our algorithm. Figure maps the locations of the core and peripheral airports. The three largest core-periphery pairs labeled 1, 2, and 3 are mainly based in Europe, East Asia, and the United States, respectively. The core-periphery pairs 1, 2, and 3 consist of the airports in 125, 35, and 47 countries, respectively. Each of the other core-periphery pairs labeled 4–10 consists of the airports in one country. The location of the airports and metropolises in Europe, East Asia, the United States and their surroundings are shown in Fig. . Here the metropolis is defined as the capital city of all countries, the provincial capitals of China and the state capitals of the United States because China and the United States have many airports. Core-periphery pair 1 contains 333 core airports and 378 peripheral airports, of which 405 (57%) airports are located in Europe [Fig. ]. However, this core-periphery pair excludes most airports in the Nordic countries (84 airports; 68%). There are 89 airports within 20 miles from a metropolis in Europe, among which there are 51 core airports (57%), 28 peripheral airports (31%), and 10 residual airports (11%). As a comparison, if we select the same number of the European airports with the largest degrees as that of the European core airports, then 64 airports (72%) are contained in the set of 89 airports within 20 miles from a metropolis, which is more than the number of the core airports (51 airports; see above) contained in the same set of airports. This result indicates that hub metropolitan airports, which are common, are not necessarily core airports. Location of the airports in (a) Europe, (b) East Asia, and (c) the contiguous United States and their surrounding areas. The large and small filled circles represent the core and the peripheral airports, respectively. Each color represents a core-periphery pair. The open squares represent residual airports. The inverted triangles indicate the location of metropolises, i.e., the capital cities of all countries, the provincial capitals of China and the state capitals of the United States. The second core-periphery pair contains 161 core airports and 240 peripheral airports, among which 217 (54%) airports are located in East Asia [Fig. ]. In this core-periphery pair, 31 airports are located within 20 miles from a metropolis in East Asia, among which there are 23 core airports (74%), eight peripheral airports, and no residual airport [Fig. ]. The third core-periphery pair contains 150 core airports and 312 peripheral airports, among which 210 (45%) airports are located in the United States [Fig. ]. In this core-periphery pair, 71 airports are located within 20 miles from a metropolis in the United States, among which there are 29 core airports (41%), 30 peripheral airports (42%), and 12 residual airports (17%) [Fig. ]. We have not found the partitioning of airports into core-periphery pairs corresponding to different major airline groups (e.g., American Airlines, Delta Airlines, Southwest Airlines, and United Airlines in the United States). Table lists the size of core-periphery pairs and the fractions of different types of edges. The airports in a large core are not densely interconnected compared to those in small core-periphery pairs, probably due to the limited capacity of the airports (e.g., a small number of runways). Core-periphery pairs 1, 2, and 3 contain hub airports in each region. The other small core-periphery pairs consist of a small number of core airports, i.e., at most 20% of the airports in each core-periphery pair. In these core-periphery pairs, most of the peripheral airports are adjacent to the core airports but not to other peripheral airports in the same core-periphery pair. This observation suggests that a small number of core airports relays most of the flights into these regions as gateway airports. For example, the representative core airport (i.e., the core airport that has the largest number of neighbors in the core-periphery pair) in pair 4, MNL (Philippines), and that in pair 8, LOS (Nigeria), serve most of the domestic airports in the respective countries. Such a structure is evident in small core-periphery pairs such as core-periphery pairs 4–10. The subnetwork within the Philippines is shown in Fig. ; see Table for properties of all airports in the Philippines. Most of the airports (34 airports; 92%) in core-periphery pair 4 [shown in orange in Figs. , , and ] only serve domestic flights. Core airport 1 [labelled in Fig. ] has most of the edges (41 edges; 84%) between core-periphery pair 4 and the rest of the network. Therefore, core airport 1 functions as a gateway airport in the Philippines. Core airport 2 also functions as a gateway airport but to a lesser extent than core airport 1 does. Core-periphery pairs located in Alaska (core-periphery pair 6 in Table ), Russia (pair 7), and Ecuador (pair 9) also contain a few core airports serving as gateway airports in the respective regions (Appendix ). Core airports 8 and 21 in the Philippines [Fig. ] have a small degree, which is counterintuitive. Core nodes having degree one or two are also found in core-periphery pair 6 [Fig. ]. The airports 8 and 21 in the Philippines are adjacent to one peripheral airport 7 and 4, respectively. If we assign airport 8 to the periphery, then two peripheral airports 7 and 8 would be adjacent. Similarly, if we assign airport 21 to the periphery, then two peripheral airports 4 and 21 would be adjacent. To avoid edges between peripheral nodes, our algorithm has identified airports 8 and 21 as core nodes. However, airports 8 and 21 may be better regarded as peripheral airports given that they are not densely interconnected to other core airports. Previous studies provided remedies for this issue (see Sec. for further discussion). Airport network within (a) the Philippines and (b) Thailand. The line color indicates the core-periphery pair to which the two airports belong. The edges connecting two airports in different core-periphery pairs are shown in gray. The numbers attached to some airports indicate the IDs of the airports listed in Tables and . We only show the IDs of all core airports, some peripheral airports and all residual airports. The subnetwork within Thailand is shown in Fig. ; see Table for properties of all airports in Thailand. Two major airports 1 and 14 are located in the capital city, Bangkok, and belong to different core-periphery pairs. All international airports in Thailand belong to core-periphery pair 2 [shown in blue in Figs. , , and ], including core airport 14. Most of the domestic airports (13 airports; 59%) belong to core-periphery pair 10 (shown in magenta), including core airport 1. The subnetwork composed of core-periphery pair 10 is largely separated from the other airports in Thailand, which belong to core-periphery pair 2, and the rest of the world. The separation of the domestic and international airports and their respective subnetworks is also observed in the Philippines [Fig. ], Iran, and Nigeria (Appendix ). We implement the three algorithms in MATLAB and run simulations on a computer with Intel 2.6-GHz Sandy Bridge processors and 4 GB of memory. The speed of an algorithm is measured by averaging the CPU time over 100 runs. We do not run the statistical test because it is a common process for the three algorithms. The average CPU time of the three algorithms is compared in Table . The BE-KL algorithm is the fastest on all synthetic networks and the karate club network. However, it is slower than our algorithm on the blog and airport networks. The two-step algorithm is the slowest on all but one network. Our algorithm is approximately two times slower than the BE-KL algorithm on the synthetic and karate club networks. However, on the blog and airport networks, it runs much faster than the BE-KL algorithm. Our algorithm runs in
We proposed a scalable algorithm to detect multiple core-periphery pairs in networks by maximizing a novel quality function
In the airport network, we have found several core nodes having degree one or two [e.g., airports 8 and 21 in Fig. ], which contradicts the intuition that core nodes would have a large degree. Our algorithm assigned these nodes to a core to suppress the edges between peripheral nodes. However, these nodes may be better regarded as peripheral nodes because they are adjacent to at most one core node. One remedy is to weaken the suppression of the edges between peripheral nodes . Adapting this idea to the case of multiple core-periphery structure warrants future research.
Previous studies provided algorithms to detect multiple core-periphery pairs based on community detection algorithms. Yang and Leskovec used an algorithm for detecting overlapping communities in networks . They regarded the nodes belonging to many communities as core nodes and nodes belonging to few communities as peripheral nodes. The algorithm may detect densely interconnected peripheral nodes because the detected peripheral nodes in a single core-periphery pair belong to the same community. In addition, a periphery may belong to multiple cores in these algorithms. These properties are shared by the algorithms proposed in Refs. . In contrast, our algorithm detects disjoint core-periphery pairs such that peripheral nodes are interconnected sparsely within each core-periphery pair and across different core-periphery pairs. Yan and Luo focused on a different type of structure consisting of multiple cores and a single periphery . In contrast, a core detected by our algorithm owns its exclusive periphery, including the case of an empty periphery.
We used the Erdős-Rényi random graph as the null model to define
There are several lines of possible extensions of the present work. First, we did not consider continuous versions of core-periphery structure, with which each node is assigned a strength (i.e., a coreness) value representing the belongingness of the node to a core . Continuous versions of core-periphery structure can reveal nested structure of cores (i.e., cores within a core), which discrete versions of the algorithms would not. Borgatti and Everett generalized a discrete version of core-periphery structure defined by Eq. to a continuous version by replacing binary variables
Second, we have ignored the weight and direction of edges. It is straightforward to incorporate the weight of edges by redefining
Third, our quality function,
How multiple core-periphery pairs emerge is unclear. An economic mechanism explains the emergence of a single core-periphery pair in networks . The authors considered the trade-offs between the profit obtained by connecting nodes and the cost for maintaining edges. Core-periphery structure emerges if the cost is not extremely small or large relative to the profit . Given their results, multiple core-periphery pairs may emerge when the cost of intergroup edges is significantly larger than the cost of intragroup edges. For example, in airport networks, interregional flights would be more costly than intraregional flights due to the different fuel expense and tax. Exploration of dynamical or economic mechanisms behind the formation of multiple core-periphery pairs in a network warrants future work.
We evaluated the performance of the Tunç-Verma (TV) algorithm using the synthetic networks used in Sec. . The Python code provided by the authors is not fast enough. Therefore, we reimplement their algorithm based on the original code by changing the data structure and replacing some functions by faster inbuilt functions in MATLAB. We did not change the algorithm itself including the parameters. We refer the original and reimplemented algorithms as the TV and
Figure shows the VI for the networks composed of a single core-periphery pair, whose structure is schematically shown in Fig. . The VI for the BE-KL and our algorithms is approximately equal to zero except for small
Figure shows the results for the networks composed of two core-periphery pairs, whose structure is schematically shown in Fig. . The VI for the BE-KL algorithm is large for all values of
In both types of the synthetic networks, the TV and
Next, we measure the speed of the
The divisive algorithm partitions the nodes into communities using the Louvain algorithm and then partitions the nodes in each community into core and peripheral nodes using the BE-KL algorithm. We apply the statistical test (Sec. ) to the core-periphery pairs detected by the divisive algorithm.
The VI values for the synthetic networks are shown in Fig. . For the synthetic network with a single core-periphery pair [Fig. ], the VI values are large in the entire
The core-periphery structure in the karate club detected by the divisive algorithm is shown in Fig. . The divisive algorithm detects two core-periphery pairs, each of which mostly consists of the members supporting the same leader [Fig. ]. The average density of intracore edges and that of core-periphery edges within a core-periphery are 1.000 and 0.619, respectively, which are larger than the edge density for the entire network,
In the blog network, the divisive algorithm detects three core-periphery pairs, each of which mostly comprises the blogs with the same political leaning [Fig. ]. Two core-periphery pairs are much larger than the third one and have the opposite political leanings. The divisive algorithm identifies more residual nodes than the two-step algorithm. The average density of intracore edges and that of core-periphery edges within a core-periphery pair are 0.3964 and 0.2906, respectively, which are larger than the edge density for the entire network,
In the airport network, the divisive algorithm identifies 12 core-periphery pairs, each of which mostly consists of the airports located in the same geographical region [Fig. ]. The average density of intracore edges and that of core-periphery edges within a core-periphery pair are 0.6335 and 0.2978, respectively, which are larger than the edge density for the entire network,
Our algorithm separates the international and domestic airports in the Philippines, Iran, and Nigeria into different core-periphery pairs. In Iran, the airports mainly serving the international and domestic flights belong to core-periphery pairs 1 and 5, respectively (Table ). In Nigeria, the airports mainly serving the international and domestic flights belong to core-periphery pairs 1 and 8, respectively (Table ). There is no clear geographical division of the international and domestic airports in Iran and Nigeria [Figs. and ].
Airport network within (a) Iran, (b) Nigeria, (c) core-periphery pair 6 based in Alaska, (d) Ecuador, and (e) Russia. The line color indicates the core-periphery pair to which the two airports belong. The edges connecting two airports in different core-periphery pairs are shown in gray. The numbers attached to some airports indicate the IDs of the airports listed in Tables –. We only show the IDs of the core airports, some peripheral airports and some residual airports.
Some core airports in Alaska, Ecuador, and Russia serve as gateway airports in the respective regions. In core-periphery pair 6 based in Alaska, core airport 1 is adjacent to all the other airports in this core-periphery pair [Fig. ]. In addition, only core airport 1 has an edge to the rest of the network (Table ) and therefore is the unique gateway airport for this core-periphery pair. In Ecuador, most of the airports (10 airports; 83%) are adjacent to airport 1, which is the unique core airport in core-periphery pair 9 [Fig. ]. This core airport has most of the edges (10 edges; 77%) between core-periphery pair 9 and the rest of the network (Table ). Therefore, core airport 1 serves as a gateway airport in Ecuador. Airport 11 also functions as a gateway airport in Ecuador. The Russian airports belong to core-periphery pair 1, 2, or 7 [Fig. ]. Most of the airports in core-periphery 7 are located in Russian Far East. In core-periphery 7, all peripheral airports are adjacent to core airport 1. The core airport 1 has most of the edges (eight edges; 67%) between core-periphery pair 7 and the rest of the network (Table ). Therefore, core airport 1 serves as a gateway airport for this core-periphery pair. There is no clear separation between the domestic and international airports into different core-periphery pairs in Russia.
References (64)
- M. E. J. Newman, Networks: An Introduction (Oxford University Press, Oxford, 2010).
- A.-L. Barabási, Network Science (Cambridge University Press, Cambridge, 2016).
- S. Fortunato, Phys. Rep. 486, 75 (2010).
- M. Girvan and M. E. J. Newman, Proc. Natl. Acad. Sci. USA 99, 7821 (2002).
- M. E. J. Newman and M. Girvan, Phys. Rev. E 69, 026113 (2004).
- L. A. Adamic and N. Glance, in Proceedings of the 3rd International Workshop on Link Discovery (ACM, New York, 2005), pp. 36–43.
- R. Guimerà, S. Mossa, A. Turtschi, and L. A. N. Amaral, Proc. Natl. Acad. Sci. USA 102, 7794 (2005).
- M. Sales-Pardo, R. Guimerà, A. A. Moreira, and L. A. N. Amaral, Proc. Natl. Acad. Sci. USA 104, 15224 (2007).
- B. Karrer and M. E. J. Newman, Phys. Rev. E 83, 016107 (2011).
- R. Guimerà and L. A. N. Amaral, Nature 433, 895 (2005).
- S. P. Borgatti and M. G. Everett, Soc. Netw. 21, 375 (2000).
- P. Holme, Phys. Rev. E 72, 046111 (2005).
- J. P. Boyd, W. J. Fitzgerald, M. C. Mahutga, and D. A. Smith, Soc. Netw. 32, 125 (2010).
- P. Csermely, A. London, L.-Y. Wu, and B. Uzzi, J. Complex Netw. 1, 93 (2013).
- S. H. Lee, M. Cucuringu, and M. A. Porter, Phys. Rev. E 89, 032810 (2014).
- M. P. Rombach, M. A. Porter, J. H. Fowler, and P. J. Mucha, SIAM J. Appl. Math. 74, 167 (2014).
- J. Yang and J. Leskovec, Proc. IEEE 102, 1892 (2014).
- A. Ma and R. J. Mondragón, PLOS ONE 10, e0119678 (2015).
- B. Tunç and R. Verma, PLOS ONE 10, e0143133 (2015).
- X. Zhang, T. Martin, and M. E. J. Newman, Phys. Rev. E 91, 032803 (2015).
- M. Cucuringu, P. Rombach, S. H. Lee, and M. A. Porter, Eur. J. Appl. Math. 27, 846 (2016).
- J. Gamble, H. Chintakunta, A. Wilkerson, and H. Krim, IEEE Trans. Signal Inf. Process. Netw. 2, 186 (2016).
- T. Verma, F. Russmann, N. A. M. Araújo, J. Nagler, and H. J. Herrmann, Nat. Commun. 7, 10441 (2016).
- D. S. Bassett, N. F. Wymbs, M. P. Rombach, M. A. Porter, P. J. Mucha, and S. T. Grafton, PLOS Comput. Biol. 9, e1003617 (2013).
- M. R. da Silva, H. Ma, and A.-P. Zeng, Proc. IEEE 96, 1411 (2008).
- S. Bruckner, F. Hüffner, and C. Komusiewicz, Algo. Mol. Biol. 10, 16 (2015).
- F. D. Rossa, F. Dercole, and C. Piccardi, Sci. Rep. 3, 1467 (2013).
- B. Craig and G. von Peter, J. Financ. Intermed. 23, 322 (2014).
- D. Fricke and T. Lux, Comput. Econ. 45, 359 (2014).
- B. Yan and J. Luo, Preprint arXiv:1605.03286 (2016).
- D. Sardana and R. Bhatnagar, in Proceedings of the 2016 IEEE/WIC/ACM International Conference on Web Intelligence (IEEE, Nebraska, 2016), pp. 1–8.
- B.-B. Xiang, Z.-K. Bao, C. Ma, X.-Y. Zhang, H.-S. Chen, and H.-F. Zhang, (2016), preprint arXiv:1612.01704 (2016).
- J. I. Alvarez-Hamelin, L. Dall'Asta, A. Barrat, and A. Vespignani, in Advances in Neural Information Processing Systems 18 (MIT Press, Cambridge, 2005), pp. 41–50.
- J. Cohen, Comput. Sci. Eng. 11, 29 (2009).
- Y. Zhao, E. Levina, and J. Zhu, Proc. Natl. Acad. Sci. USA 108, 7321 (2011).
- J. Chen and Y. Saad, IEEE Trans. Knowl. Data Eng. 24, 1216 (2012).
- P. Erdős and A. Rényi, Publ. Math. Inst. Hung. Acad. Sci. 5, 17 (1960).
- J. P. Boyd, W. J. Fitzgerald, and R. J. Beck, Soc. Netw. 28, 165 (2006).
- D. in 't Veld and I. van Lelyveld, J. Bank. Finac. 49, 27 (2014).
- U. N. Raghavan, R. Albert, and S. Kumara, Phys. Rev. E 76, 036106 (2007).
- V. D. Blondel, J.-L. Guillaume, R. Lambiotte, and E. Lefebvre, J. Stat. Mech. (2008) P10008.
- B. W. Kernighan and S. Lin, Bell Syst. Tech. J. 49, 291 (1970).
- Z. Šidák, J. American Stat. Assoc. 62, 626 (1967).
- M. Meilă, J. Multivariate Anal. 98, 873 (2007).
- C. O. Flores, S. Valverde, and J. S. Weitz, ISME J. 7, 520 (2013).
- T. P. Peixoto, Phys. Rev. E 95, 012317 (2017).
- W. W. Zachary, J. Anthropol. Res. 33, 452 (1977).
- J. Patokallio, “Openflights data” (2009) [http://openflights.org].
- T. Opsahl, “Why Anchorage is not (that) important: Binary ties and sample selection” (2011) [https://toreopsahl.com/2011/08/12/why-anchorage-is-not-that-important-binary-ties-and-sample-selection].
- J. Yang and J. Leskovec, in Proceedings of the 2012 IEEE 12th International Conference on Data Mining (IEEE, Washington, 2012), pp. 1170–1175.
- J. Yang and J. Leskovec, in Proceedings of the Sixth ACM International Conference on Web Search and Data Mining (ACM, New York, 2013), pp. 587–596.
- J. Reichardt and S. Bornholdt, Phys. Rev. Lett. 93, 218701 (2004).
- J. Reichardt and S. Bornholdt, Phys. Rev. E 74, 016110 (2006).
- R. Mastrandrea, T. Squartini, G. Fagiolo, and D. Garlaschelli, New J. Phys. 16, 043022 (2014).
- V. A. Traag and J. Bruggeman, Phys. Rev. E 80, 036115 (2009).
- M. MacMahon and D. Garlaschelli, Phys. Rev. X 5, 021006 (2015).
- M. J. Barber, Phys. Rev. E 76, 066102 (2007).
- P. Expert, T. S. Evans, V. D. Blondel, and R. Lambiotte, Proc. Natl. Acad. Sci. USA 108, 7663 (2011).
- A. Arenas, J. Duch, A. Fernández, and S. Gómez, New J. Phys. 9, 176 (2007).
- E. A. Leicht and M. E. J. Newman, Phys. Rev. Lett. 100, 118703 (2008).
- Y. Kim, S.-W. Son, and H. Jeong, Phys. Rev. E 81, 016103 (2010).
- M. E. J. Newman, Phys. Rev. E 94, 052315 (2016).
- S. Fortunato and M. Barthélemy, Proc. Natl. Acad. Sci. USA 104, 36 (2006).
- R. Guimerà, M. Sales-Pardo, and L. A. N. Amaral, Phys. Rev. E 70, 025101 (2004).